Credit hybrids can be considered to be financial instruments with at least two sources of risk. Typically, one of those risks is credit risk, such as the risk that a party will experience an adverse credit event, such as insolvency or defaulting on a loan or bond obligation. Other risks that may be implicated in a credit hybrid include interest rate risk and foreign exchange risk. Many such credit hybrids are difficult to price because there typically is not a liquid market for such instruments. Nevertheless, a party owning such credit hybrids has an interest in pricing them, particularly when determining how to hedge the instruments.
Because there typically is no liquid market for such credit hybrids, they are often priced using mathematical models. It is known that the price of an instrument that is dependent upon the credit risk of a party is related to the probability of survivability of that party. The probability of survivability of the party, denoted Psurv, at the end of future time τ, measured at time t can be expressed as:Psurv(t)=e−∫0h(s,s)ds   (1)where h(t,t) is the instantaneous hazard rate. In the past, it was known to model the forward hazard rate with the following stochastic equation:dhtT=μtThdt+σtThdBth, Bth∈R1   (2)where T is the maturity of the corresponding hazard rate, μtThdt is the hazard rate drift term, σtTh is the hazard rate volatility, and Bth is the continuous-time Brownian motion. A general discussion of this model is contained in P. J. Schonbucher, “Term structure modeling of defaultable bonds,” The Review of Derivatives Studies, Special Issue: Credit Risk, 2(2/3): 161-192, 1998. While this model is valuable, it does not address any issues related to a proper treatment of the credit volatility skew, such as the probability of credit spreads going up is greater than the probability spreads going down and vice versa.